Good reduction and cyclic covers

TL;DR

This paper establishes finiteness results for varieties over number fields with controlled bad reduction, using cyclic covers and moduli stacks, advancing the understanding of the Shafarevich conjecture in specific geometric contexts.

Contribution

It proves new finiteness theorems for varieties with good reduction outside a finite set, applying cyclic covers and stack techniques to generalize the Shafarevich conjecture.

Findings

Finiteness results for weighted projective surfaces

Finiteness for double covers of abelian varieties

Reduction of the Shafarevich conjecture for hypersurfaces

Abstract

We prove finiteness results for sets of varieties over number fields with good reduction outside a given finite set of places using cyclic covers. We obtain a version of the Shafarevich conjecture for weighted projective surfaces, double covers of abelian varieties, and reduce the Shafarevich conjecture for hypersurfaces to the case of hypersurfaces of high dimension. These are special cases of a general set-up for integral points on moduli stacks of cyclic covers, and our arithmetic results are achieved via a version of the Chevalley-Weil theorem for stacks.